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SAAEI 2014. Tangier, 25-27 June. CDES-6 A Harmonic Balance Approach for Bifurcation Analysis of a Ferroresonant Circuit J. A. COREA-ARAUJO, A. EL AROUDI, F. GONZÁLEZ-MOLINA, J. A. MARTÍNEZ-VELASCO, J. A. BARRADO-RODRIGO, AND L. GUASCH-PESQUER Abstract—Despite the extensive available literature on this topic, ferroresonance prediction and behavior characterization still remain widely unknown. The Harmonic Balance Method is presented in this paper as an approach to predict the dynamics of a ferroresonance system. The implication of initial conditions in the final ferroresonant state is analyzed. Bifurcation diagrams are obtained to depict the path in which the phenomenon will move and to predict the critical points where ferroresonance occurs. Stability region for the test system is also located from the study. Keywords: Ferroresonance, Harmonic Balance, Bifurcation, Stability, Transformers. F I. INTRODUCTION ERRORESONANCE is one of the most common transient and steady-state phenomena studied by power system engineers. The phenomenon usually appears in scenarios involving a non-linear inductance associated with any capacitance source. Such interaction unfolds not only into potentially dangerous overvoltages, jumping to a high current fundamental-frequency state but also bifurcations to subharmonic, quasi-periodic and even chaotic oscillations. The phenomenon is commonly initiated after some type of switching event such as load rejection, fault clearing, transformer energization, single-phase switching or loss of system grounding. In general, ferroresonance can occur in any power system that contains transformers, regardless its type or size. Capacitance interaction can be found either in the form of actual capacitor banks or capacitive coupling [1]. Although ferroresonance has been present in power systems for more than a century, and despite the extensive literature available nowadays, its behavior and characterization still remain widely unknown [2]. The first analytical works were presented in the 50’s [3], [4]. In the 70’s and 80’s, digital simulation began to replace transient network analyzers (TNA) following pioneering work in digital computer solution methods by Dommel [5]. After the breakthrough of the nonlinear dynamics and chaos theory in the late 70’s, a new analysis door was open. The Javier A. Corea-Araujo, Abdelali El Aroudi, Francisco González-Molina, José A. Barrado-Rodrigo, and Luis Guasch-Pesquer are with the Departament d'Enginyeria Electrònica, Elèctrica i Automàtica, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]) Juan A. Martinez is with the Departament d'Enginyeria Elèctrica, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain. (e-mail: [email protected]) connection of ferroresonance to nonlinear dynamics and chaos allowed to increase the expertise in waveform analysis and enabled a classification of ferroresonance states into different modes [6]: Fundamental mode (voltages and currents are periodic with a period T equal to the power frequency period), sub-harmonic mode (periodic signals with a period multiple of the source period), quasi-periodic mode (non-periodic signal, composed by two or more periodic signals with incommensurable period), and chaotic mode (signals showing an irregular and unpredictable behavior). The first work linking up ferroresonance with non-linear analysis was published in 1992 [7]. That work presented a three-phase transformer leading to a chaotic state. Topics such as bifurcation theory, global dynamic behavior or the Galerkin method were common for ferroresonance analysis during the 90’s [8],[9]. However, it was not until 2000 when several techniques obtained from the merge with nonlinear dynamics were developed [10]. The first significant contribution was presented in [11]. Research topics such as ferroresonance behavioral patterns [12], hysteresis impacts [13], transformer core studies [14] or new transformer models derivation [15], are usually followed by the words ‘stability domain’, ‘dynamic behavior’ or ‘harmonic balance’ [16]. In this research, a predictive analysis is presented to locate stability domains for a period-1 ferroresonance. This prediction is made by using the Harmonic Balance method to solve the differential equation of the system. The order of polynomial expression representing the non linearity has been strategically selected to approach the physical characteristic of the inductor. The solution technique used in the harmonic balance allows comparing the procedure with the widely studied Duffing’s equation solution [11]. The document is organized as follows. A detailed description of the system under study is presented in Section II. The implementation of the Harmonic Balance Method is presented in Section III. Validations through bifurcation diagrams obtained from system level simulations are presented in Section IV. Conclusions are given in Section V. II. TEST SYSTEM DESCRIPTION To better explain the approach of this paper, the basic circuit shown in Fig. 1, will be analyzed. The circuit is composed of a 25 kV/50 Hz power source, a capacitance of 5 µF and a saturable inductance paralleled with a resistance of 40 kΩ representing the core losses [1]. ID: 16012 III. THE HARMONIC BALANCE METHOD Fig. 1. Diagram of the test circuit A proper representation of the transformer saturation is considered vital in ferroresonance studies. Hence, when a mathematical equivalence of a transformer circuit is developed, the expression representing the saturation effect must be close-fitted to actual test data. Several approaches have been used throughout the years, being the two most accurate the trigonometric approximation based in the hyperbolic arctangent properties [17], and the polynomial approximation, whose strength relies on its simple implementation and the easy calculation of its coefficients [11], [18]. In this work, the polynomial approximation has been used due to its suitability on the harmonic balance resolution and the possibility of comparing with previous works. Expression (1) shows the polynomial form. (1) where iL represents the magnetizing current, λ represents the fluxlinked and a and b the coefficients for the curve fitting. Selecting a proper value of n in the polynomial expression will significantly mark the accuracy of representation. Figure 2 shows different curves corresponding to (1) with different n values and the actual test data. The figure proves that higher values of n approach better to the real nature of magnetization curve and its utilization will make the modeling more realistic. From Figure 2, the differential equation representing the mathematical behavior of the system is as followed: (2) cos where k= 1/RmCs, C1= a/Cs, C2= b/Cs and G=ωVrms. In order to best approach the physical reality of the saturation curve ‘n’ is set to 11. The coefficients are set as follows: 43.59 10 and 7.415 10 . The harmonic balance is a method for the study of nonlinear oscillating systems which are defined by non-linear ordinary differential equations [19]. Few approaches has been made in the ferroresonance field, most of them only consider the sinousoidal approximation or consider a low order polynomial approximation [20], [21]. In general terms, the method consists on substituting the unknown variable in the equation by an assumed solution so that approximate periodic solutions can be found. The forced solution can be selected to behave as a steady state solution using a truncated Fourier series [18]. # * $ %& cos '& sin +,# (3) In this work the dc component can be neglected due to the sinusoidal state of the source signal. Here k sets the number of harmonic components. Normally this value depends on the oscillations considered or expected. For this work, k is set to 1 to obtain a first order approximation. Thus, the equation (3) will have the form: (4) % cos ' sin In order to substitute (4) into (2), the first and second derivatives should be found. These are: - % sin ' ωcos (5) (6) -% ω cos - ' ω sin Substituting the expression for the flux and its derivatives in (2) and having the value ‘n’ as 11, will significantly change the expression (2) in size and complexity. Once simplification is performed, and the third and higher harmonics are neglected, a set of expressions are obtained for first-order solution of the parameters A and B. These are given as follows: %/ - '/ - 231 512 % ' 1 % ' 1 231 512 2 ' (7) 2-% 0 (8) A general expression can be found by using the polar solution of the flux: where % cos ' sin 3 cos 4 (9) (10) 3 5% ' Thus, r by convenience provides also the magnitude of the flux. Then, (7) and (8) merges in a single expression: 6 3 Fig. 2. Non-linear inductance characteristic 3 / - 231 512 3 #2 (11) According to [9] real values of r that satisfy (11) correspond to the periodic solutions for the original differential equation (2). A graphical explanation of (11) can be made using a 1-D bifurcation diagram. It is obtained by founding the analytically solution for (2) while a system parameter is varied (source ID: 16012 voltage). Figure 2 shows the result for the system. It can be deduced from this figure that, when the bifurcation parameter increases, the response of the flux suddenly jumps when it approaches the first turning point (TP1). The same will happen when the bifurcation parameter is slowly decreased; the flux will abruptly drop while approaching the second turning point (TP2). This type of behavior is usually referred as hysteretictype pattern and it is associated to a saddle-node bifurcation [9]. The frontier at which the jumps occur can be determined by founding the threshold of the change just when (10) has mind that the harmonic balance prediction is based on a firstorder harmonic approach, which implies that only the sinusoidal stage is considered whereas computational systems include all the harmonic series in their solution. In addition, it is important remarking that EMTP-ATP simulation can take several hours without including the post-processing task. Harmonic balance can take no more than an hour. 78 only one solution instead of two. This occurs when 0. 79 The derivative of (11) will give solution for (2) at the turning points and equating it to 0. 23 : 11 = 3 - - ; ; 1 1 3 # 3 # < 0 (12) If different capacitance values are tested, the corresponding values of voltage for the turning points can be calculated evaluating (13) for Vrms, and using the pairs (r,Cs) obtained from (12) [13]. 3 / - 231 512 3 #2 >[email protected] (13) The graphical representation of equation (13) is known as bifurcation line or stability domain boundary [11]. The 2-D bifurcation map from Figure 3 has a significant meaning for ferroresonance analysis: it allows predicting where the boundary between sinusoidal and ferroresonance response is located. For instance, every combination of values at the right of TP1 line will have a high flux peak value while left side combinations could have both low flux values for TP1 conditions and high flux values response due to the coexistence with TP2 conditions. Fig. 2. 1-D bifurcation diagram for Cs=5µF. IV. BIFURCATION DIAGRAMS FROM THE COMPUTATIONAL SYSTEM MODEL In order to benchmark the analysis, a set of bifurcation maps will be presented from the test system. Time domain simulations will be carried out under MATLAB environment to validate the mathematical analysis and the veracity in the harmonic balance approximation. Figure 4 shows the bifurcation diagram obtained from time domain simulation. This result proves the two different solutions predicted by the harmonic balance. One obtained by parametrically increasing the voltage, and the other by decreasing the same parameter. It is important to emphasize the co-existence of the two solutions, which reinforces the influence that initial conditions have over the behavior of a ferroresonant system. To confirm the result derived from the harmonic balance analysis, an EMTP simulation study has been performed. Figure 5 presents the 1-D bifurcation diagram originated from EMTP-ATP simulation. The results are significantly close to harmonic balance prediction. A small discrepancy can be detected on the upper stable solution when comparing Figure 2 against Figures 4 and 5. However, it is important to keep in Fig. 3. 2-D bifurcation diagram. Fig. 4. 1-D bifurcation diagram obtained using MATLAB for Cs=5µF. ID: 16012 V. CONCLUSIONS Fig. 5. 1-D bifurcation diagram obtained using EMTP-ATP for Cs=5µF. This paper has presented the application of the Harmonic Balance Method for ferroresonance analysis of a simplified transformer model by respecting the true nature of the magnetic saturation. The solutions were derived from a highorder polynomial so the results are close to the physical reality of a transformer. Comparisons of time-domain simulations using MATLAB and EMTP-ATP were presented. The analysis presented in this work has remarked the influence of the initial conditions in ferroresonance representation; the proper characterization can be vital for a well developed analysis. The main contribution of the paper has been the application of harmonic balance as a predictive tool. Its capability to describe the behavior tendency of a system prone to ferroresonance makes it a powerful technique. At the same time its simplicity makes it suitable to substitute large parametric analysis avoiding extensive time-domain simulations. Finally, bifurcation maps resulting from harmonic balance analysis provides clear paths for system design that could avoid hypothetical ferroresonance conditions. VI. ACKNOWLEDGMENT This work is being supported by the Spanish “Ministerio de Educación y Ciencia” under the Grant DPI2012-31580. VII. REFERENCES Fig. 6. System response for Vrms=25 kV and Cs=10 µF. Fig. 7. System response for Vrms=9 kV and Cs=10 µF . From Figure 3 it is easy to differentiate the regions for the stability of the systems. Any match of parameters selected at the right of the TP1 boundary will result into an undesirable solution; that is, in a ferroresonant state. A time-domain simulation has been performed to prove the previous statement. Figure 6 validates the prediction for the undesirable region. However, the region at the left of TP1 in Figure 3 is totally dependent of the initial condition on the system. Two different stages could occur. On one hand, the system could work in a sinusoidal state, normally for initial conditions set to zero. 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